The Form of Prime Numbers

Brian Stedjee, June 8. 2014

All prime numbers except for 2 and 3 are of the form 6n + 1 or 6n – 1 where n is a positive integer. A positive integer is a prime if it is not divisible by 2, 3, or another prime number of the form 6n ± 1.

No number of the form 6n ± 1 is divisible by 2 because all numbers divisible by 2 are even and all numbers of the form 6n ± 1 are odd. Also, no numbers of the form 6n ± 1 are divisible by 3. The fact that all numbers of the form 6n are divisible by 3 eliminates the possibility that any number 6n ± 1 or, or that matter, 6n ± 2 is divisible by 3.

We may conclude then that all positive integers of the form 6n ± 1 are prime when they are not the product of the primes in the form 6n ± 1. that is, all positive integers are prime when they do not equal (6A ± 1)(6B ± 1) where A and B are positive integers. In addition, no positive integer of the form 6n ± 1 is prime when it does equal (6A ± 1)(6B ± 1) where A and B are positive integers.

Non-prime integers of the form 6n ± 1 then are equal to the product of (6A + 1)(6B + 1) or (6A – 1)(6B – 1). Expanding…

6n + 1 = 36AB + 6A + 6B +1

or

6n + 1 = 36AB – 6A – 6B +1

6n = 36AB + 6A + 6B

6n = 36AB – 6A – 6B

n = 6AB + A + B

and when 6n + 1 isn’t prime.

n = 6AB – A – B

Non-prime integers of the form 6n – 1 are equal to the product of (6A + 1)(6B – 1) or

(6A -1)(6B +1) so…

n = 6AB + A – B

and when 6n - 1 isn’t prime.

n = 6AB – A + B

THEOREMS:

1. If a positive integer (n) cannot be generated by the formula 6AB ± A ± B = n, where A and B are positive integers, then 6n + 1 and 6n – 1 are both prime (e.g. no combination of A and B can generate n = 1, 2, or 3)

6(1) – 1 = 5 is prime

6(1) + 1 = 7 is prime

6(2) – 1 = 11 is prime

6(2) + 1 = 13 is prime

6(3) – 1 = 17 is prime

6(3) + 1 = 19 is prime

2. If one or more combinations of A and B exist such that A’s and B’s are either both added to or subtracted to 6AB to produce n (i.e. 6AB + A + B = n or 6AB – A – B = n) then 6n – 1 is prime while 6n + 1 is not.

examples:

6(1)(1) – 1 – 1 = 4 = n

6(4) – 1 = 23 is prime

6(4) + 1 = 25 is not prime

6(1)(6) – 1 – 6 + 29 = n

6(1)(4) + 1 + 4 = 29 = n

6(29) – 1 = 173 is prime

6(29) + 1 = 175 is not prime

3. If one or more combinations of n = 6AB ± A ± B exist such that the signs preceding A and B are opposite in all cases, then 6n +1 is prime, but 6n – 1 isn’t.

examples:

6(1)(2) + 1 – 2 = 11 = n

6(11) – 1 = 65 is not prime

6(11) + 1 = 67 is prime

6(2)(4) + 2 – 4 = 46 = n

6(1)(9) + 1 – 9 + 46 = n

6(46) – 1 – 275 is not prime

6(46) + 1 = 277 is prime

4. If two or more combinations of 6AB ± A ± B = n exist such that the signs preceding A and B are neither the same in all cases nor opposite in all cases, then neither 6n -1 nor 6n + 1 is prime.

examples:

6(2)(2) – 2 – 2 = 20 = n

6(1)(3) – 1 + 3 = 20 = n

6(20) – 1 = 119 is not prime (7)(17)

6(20) + 1 = 121 is not prime (11) (11)

6(1)(6) + 1 – 6 = 31 = n

6(2)(3) – 2 – 3 = 31 = n

6(31) – 1 = 185 is not prime

6(31) + 1 = 187 is not prime (11)(17)

5. Non-prime numbers of the form 6n ± 1 are all generated by 4 equations…

6{6(AB) – A – B} + 1

6{6(AB) + A + B} + 1

6{6(AB) – A + B} – 1

6{6(AB) + A - B} – 1

So the number of prime numbers less than a given number of possible combinations of 6n ± 1 minus all the possible combinations of the above non-prime numbers where n is less than the given number. It is therefore possible to determine the ordinal occurrences of a prime number by subtracting the number of non-prime numbers (above) of the form 6n ± 1 from the total number of possible combinations of 6n ± 1

6. A number of the form 6n ± 1 can be determined by its ordinal occurrence (x) by the formulas:

1 + 3(x – 2) if x is even

2 + 3(x – 2) if x is odd

Hence the 5^th prime number is 2 + 3(5-2) = 11. Therefore a prime number may be generated from its ordinal occurrence by the formula:

1 + 3[(x + N) – 2] if x + N is even

or

2 + 3[(x + N) – 2] if x + N is odd

where N is the number of non-primes of the form 6n ± less than the desired prime number.

Hence the twelfth prime number is:

1 + 3[(12 + 2) – 2] = 37

The two n’s less than 37 in this case are 25 or 6[6(1) – 1 – 1] + 1 and 35 or 6[6(1) – 1 + 1] – 1.

The principal problem then, of generating a prime number from its ordinal occurrence, is finding the number of non-prime numbers of the form 6n ± 1 less than that prime number.

These could be found by computer, by substituting all possible integers for A and B into the formulas: 6[6(AB) + A + B] +1, 6{6(AB) –A –B] +1 and 6[6(AB) 1 A + B] – 1.

1: 6[6(1) – 1 – 1] + 1 = 25

2: 6[6(1) – 1 + 1] – 1 = 35

3: 6(6(1) + 1 + 1] + 1 = 49

4: 6(6(2) – 1 – 2] + 1 = 55

5: 6[6(2) + 1 – 2] – 1 = 65

6: 6[6(2) – 1 + 2] – 1 = 77

The pattern to prime numbers, then, is really the pattern of all 6n ± 1 numbers, minus the patterns generated by the 4 equations that produce non-prime 6n ± 1. i.e.:

36AB – 6A + 6B + 1 = n

36AB + 6A + 6B + 1 = n

36AB + 6A – 6B – 1 = n

36AB – tA + 6B – 1 = n

And there are patterns to these equations. For example, in the equation 36AB – 6A – 6B + 1 = n, when A = 1, n = 25, 55, 85, 115, etc. When A = 2, n = 55, 121, 187, 253, etc.

Note that in the case where A = 1, each n is different by 30. When A = 2, each n is different by 66. All the equations that generate non-prime, 6n ± 1 numbers generate numbers that differ by a set amount, as they do in the examples above.

6n ± 1 numbers minus 36AB ± 6A ± 1 products of prime

5

7

11

13

17

19

23

25 [(6)(1) – 1] [(6)(1) – 1] (5)(5)

29

31

35 [(6)(1) + 1)] [(6)(1) – 1] (7)(5)

37

41

43

47

49 [(6)(1) + 1] [(6)(1) + 1] (7)(7)

53

55 [(6)(2) – 1] [(6)(1) + 1] (11)(7)

59

61

65 [(6)(2) + 1] [(6)(1) – 1] (13)(5)

67

71

73

77 [(6)(2) – 1] [(6)(1) + 1] (11)(7)

79

83

85 [(6)(3) – 1] [(6)(1) + 1] (17)(5)

89

91 [(6)(2) + 1] [(6)(1) + 1) (13)(7)